Fixed \(n\in \mathbb {N}\) .
By Proposition REF , for arbitrary \(z^1,z^2\in \operatorname{graph}g_{0}\)  we have
\(\dfrac{\det ^{\prime }(D\widetilde{f}^n_{-n}(z^1_{-n})|E^1_{-n})}{\det ^{\prime }(D\widetilde{f}^n_{-n}(z^2_{-n})|E^2_{-n})}=\prod _{k=1}^{n} \dfrac{\det ^{\prime }(D\widetilde{f}_{-k}(z^1_{-k})|E^1_{-k})}{\det ^{\prime }(D\widetilde{f}_{-k}(z^2_{-k})|E^2_{-k})}.\) 
We will use the following refinement version of Proposition REF , which emphasizes the dependence of the constants and can be referred to [1]}.
Lemma 4.5 ([1]} Lemma 6.10) 
For any \(m\in \mathbb {N}\) , there is a constant \(C_m>1\)  with the following property.
Let \((\mathcal {B}_1,|\cdot |_1)\) , \((\mathcal {B}_2,|\cdot |_2)\)  be Banach spaces and fixed \(M>1\) .
Assume \(A_1,A_2:\mathcal {B}\rightarrow \mathcal {B}\)  are two injective bounded linear operators and \(E_1,E_2\)  are \(m\) -dimensional subspaces of \(\mathcal {B}_1\)  satisfied that
\(|A_j|&, |(A_j|_{E_j})^{-1}|\le M \quad \text{for} \ j=1,2, \\|A_1-A_2|&, d_H(E_1,E_2)\le \dfrac{1}{C_m M^{10m}},\) 

then, we have
\(|\log \dfrac{\det (A_1|_{E_1})}{\det (A_2|_{E_2})}|\le C_m M^{10m}(|A_1-A_2|+d_H(E_1,E_2)).\) 

Let \(m_u:=\dim E^{u}(\omega ,x)\) .
For each fixed \(1\le k\le n\) , we choose \(M_k\)  so that \(D\widetilde{f}_{-k}(z^i_{-k})\) , \(E^i_{-k}\) , \(i=1,2\)  and \(C_{m_u}\)  satisfy the hypotheses of Lemma REF . We need the following:
\(&|D\widetilde{f}_{-k}(z^i_{-k})|^{\prime }_{},|(D\widetilde{f}_{-k}(z^i_{-k})|_{E^i_{-k}})^{-1}|^{\prime }_{}\le M_k \ \text{for} \ i=1,2,  \\&|D\widetilde{f}_{-k}(z^1_{-k})-D\widetilde{f}_{-k}(z^2_{-k})|^{\prime }_{}, d^{\prime }_H(E^1_{-k},E^2_{-k})\le C_{m_u} M^{10m_u},\) 

where \(d^{\prime }_H\)  refers to the Hausdorff distance in the adapted norm \(|\cdot |^{\prime }_{-k}\) .
First we choose \(M_k\)  so that \((\ref {eq:111})\)  holds.
Note that \(|(D\widetilde{f}_{-k}(z^i_{-k})|_{E^i_{-k}})^{-1}|^{\prime }_{}\le 1\)  for \(i=1,2\) .
By the choice of the function \(\ell (\omega ,x)\) , for \(i=1,2\)  we have
\(|D\widetilde{f}_{-k}(z^i_{-k})|^{\prime }_{}\le 3\ell (-k)\cdot |(f_{\theta ^{-k}\omega }|_{B(A_{\theta ^{-k}\omega },r_0)})|_{C^2}\le 3e^{2k\varepsilon _1}\ell (0)^{2}.\) 
So, on setting \(M_k:=3e^{2k\varepsilon _1}\ell (0)^{2}\) , \((\ref {eq:111})\)  satisfied.
Next, we estimate the two terms on \((\ref {eq:222})\) .
By Lemma REF  and Theorem \(\ref {Thm:Unstable}\)  (3) we have
\(|D\widetilde{f}_{-k}(z^1_{-k})-D\widetilde{f}_{-k}(z^2_{-k})|^{\prime }_{}&\le \ell (k) \cdot |z^1_{-k}-z^2_{-k}|^{\prime }_{-k}\\&\le (\dfrac{e^{\varepsilon _1}}{e^{\lambda }-\delta _1})^k\cdot \ell (0) \cdot |z^1-z^2|^{\prime }_{0}.\) 

By a simple computation and Theorem \(\ref {Thm:Unstable}\)  (2),
\(d^{\prime }_H(E^1_{-k},E^2_{-k})&\le 2|Dg_{-k}(u^1_{-k})-Dg_{-k}(u^2_{-k})|^{\prime }\\&\le 2C_0\ell (-k)\cdot |u^1_{-k}-u^2_{-k}|^{\prime }_{-k}\\&= 2C_0\ell (-k)\cdot |z^1_{-k}-z^2_{-k}|^{\prime }_{-k}\\&\le (\dfrac{e^{\varepsilon _1}}{e^{\lambda }-\delta _1})^k\cdot 2C_0\ell (0) \cdot |z^1-z^2|^{\prime }_{0},\) 

where \(z^i_{-k}=u^i_{-k}+g_{-k}(u^i_{-k})\)  for some \(u^i_{-k}\in E^u(\Phi ^{-k}(\omega ,x))\)  and \(i=1,2\) .
To fulfill the inequality \((\ref {eq:222})\) , we take \(|z^1-z^2|^{\prime }_{0}\)  small enough so that
\(\max \lbrace 2C_0,1\rbrace (\dfrac{e^{\varepsilon _1}}{e^{\lambda }-\delta _1})^k\cdot \ell (0) \cdot |z^1-z^2|^{\prime }_{0}\le ( C_{m_u} M^{10m_u}_{k})^{-1}.\) 
That is,
\(|z^1-z^2|^{\prime }_{0}\le (3^{10m_u}C_{m_u}\ell (0)^{20m_u+1} \max \lbrace 2C_0,1\rbrace (\dfrac{e^{(20m_u+1)\varepsilon _1}}{e^{\lambda }-\delta ^{\prime }_1})^k)^{-1}.\) 
By the choice of \(\varepsilon _1\ll \lambda \)  one may assume \(e^{(20m_u+1)\varepsilon _1}<e^{\lambda }-\delta ^{\prime }_1\) .
So, we take \(D^{\prime }_{\ell (0)}\)  large enough for which
\(D^{\prime }_{\ell (0)}\ge \delta ^{\prime }_1 3^{10m_u}C_{m_u}\ell (0)^{20m_u+1} \max \lbrace 2C_0,1\rbrace ,\) 
one has () holds for every \(z^1,z^2\in \operatorname{graph}g_{0}\)  with \(|z^1-z^2|^{\prime }_{0}\le \delta ^{\prime }_1 (D^{\prime }_{\ell })^{-1}\) .
Therefore, by Lemma REF  for every \(z^1,z^2\in \operatorname{graph}g_{0}\)  with \(|z^1-z^2|^{\prime }_{0}\le \delta ^{\prime }_1 (D^{\prime }_{\ell (0)})^{-1}\)  and every \(1\le k\le n\)  we have
\(\begin{aligned}\bigg | \log \dfrac{\det ^{\prime }(D\widetilde{f}_{-k}(z^1_{-k})|E^1_{-k})}{\det ^{\prime }(D\widetilde{f}_{-k}(z^2_{-k})|E^2_{-k})}\bigg |\le K^{\prime } \ell (0)^{20m_u+1} |z^1-z^2|^{\prime }_{0}(\frac{e^{(20m_u+1)\varepsilon _1}}{e^{\lambda }-\delta ^{\prime }_1})^k,\end{aligned}\) 
where \(K^{\prime }:=C_{m_u}3^{10m_u}(2C_0+1)\) .
Together with (REF ), one has
\(\bigg | \log \dfrac{\det ^{\prime }(D\widetilde{f}^n_{-n}(z^1_{-n})|E^1_{-n})}{\det ^{\prime }(D\widetilde{f}^n_{-n}(z^2_{-n})|E^2_{-n})} \bigg | &\le K^{\prime } \ell (0)^{20m_u+1} |z^1-z^2|^{\prime }_{0} \sum _{k=1}^{n}(\frac{e^{(20m_u+1)\varepsilon _1}}{e^{\lambda }-\delta ^{\prime }_1})^k\\&\le K^{\prime \prime }\ell (0)^{20m_u+1}|z^1-z^2|^{\prime }_{0},\) 

where \(K^{\prime \prime }\)  is independent of \(n\)  and \((\omega ,x)\) .
By increasing \(D^{\prime }_{\ell (0)}\)  once more so that \(D^{\prime }_{\ell (0)}\ge K^{\prime }\ell (0)^{20m_u+1}\) , the statement of Lemma REF  follows.
